Recognizing certain rational homogeneous manifolds of Picard number 1 from their varieties of minimal rational tangents.

*(English)*Zbl 1182.14042
Lau, Ka-Sing (ed.) et al., Third international congress of Chinese mathematicians. Part 1. Proceedings of the ICCM ’04, Hong Kong, China, December 17–22, 2004. Providence, RI: American Mathematical Society (AMS); Somerville, MA: International Press (ISBN 978-0-8218-4454-0/pbk; 978-0-8218-4416-8/set). AMS/IP Studies in Advanced Mathematics 42, 1, 41-61 (2008).

Let \(X\) be a complex projective manifold that is uniruled, i.e., there exists a family of rational curves \(f: \mathbb P^1 \rightarrow X\) that dominates \(X\). For a given polarization a minimal rational curve is a rational curve that belongs to such a dominating family and whose degree with respect to the polarization is minimal. If we fix a general point \(x \in X\), the minimal rational curves passing through \(x\) are parametrized by a closed subset \(\mathcal K_x\) of the Hilbert scheme. There is a natural rational map \(\mathcal K_x \dashrightarrow \mathbb P(T_{X,x})\) which associates to a curve its tangent direction at the marked point \(x\). The variety of minimal rational tangents \(\mathcal C_x\) is defined as the strict transform of \(\mathcal K_x\) under this rational map. The geometric study of varieties of minimal rational tangents (VMRTs) is guided by the idea that one should be able to recover the geometry of \(X\) from \(\mathcal C_x\), at least when \(X\) is a Fano manifold with Picard number one.

The main result of this paper confirms this idea for certain model spaces. More precisely let \(S\) be a rational homogeneous manifold with Picard number one which is either a Hermitian symmetric space or a Fano contact manifold, and denote by \(\mathcal C_0 \subset \mathbb P(T_{S,0})\) its VRMT at a point \(0 \in S\). Let \(X\) be a Fano manifold with Picard number one, and denote by \(\mathcal C_x \subset \mathbb P(T_{X,x})\) its VMRT at a general point \(x \in X\). If \(\mathcal C_0\) and \(\mathcal C_x\) are isomorphic as projective subvarieties, then \(S\) is isomorphic to \(X\). This result generalizes a famous characterization of the projective space by K. Cho, Y. Miyaoka and N. Shepherd-Barron [Adv. Stud. Pure Math. 35, 1–88 (2002; Zbl 1063.14065)]. It also generalizes an earlier result due to J.-M. Hwang and the author [J. Reine Angew. Math. 490, 55–64 (1997; Zbl 0882.22007)] where they made the stronger assumption that \(\mathcal C_0\) and \(\mathcal C_x\) are isomorphic for every point \(x \in X\). The main difficulty of the proof is that a priori there might exist a divisor \(H \subset X\) such that for \(x \in H\), the VMRT \(\mathcal C_x\) is not isomorphic to \(\mathcal C_0\). In order to exclude this possibility the author uses a notion of parallel transport along the tautological lifting of a standard minimal rational curve, a concept that already appeared implicitly in N. Mok [Trans. Am. Math. Soc. 354, No. 7, 2639–2658 (2002; Zbl 0998.32013)].

For the entire collection see [Zbl 1135.00009].

The main result of this paper confirms this idea for certain model spaces. More precisely let \(S\) be a rational homogeneous manifold with Picard number one which is either a Hermitian symmetric space or a Fano contact manifold, and denote by \(\mathcal C_0 \subset \mathbb P(T_{S,0})\) its VRMT at a point \(0 \in S\). Let \(X\) be a Fano manifold with Picard number one, and denote by \(\mathcal C_x \subset \mathbb P(T_{X,x})\) its VMRT at a general point \(x \in X\). If \(\mathcal C_0\) and \(\mathcal C_x\) are isomorphic as projective subvarieties, then \(S\) is isomorphic to \(X\). This result generalizes a famous characterization of the projective space by K. Cho, Y. Miyaoka and N. Shepherd-Barron [Adv. Stud. Pure Math. 35, 1–88 (2002; Zbl 1063.14065)]. It also generalizes an earlier result due to J.-M. Hwang and the author [J. Reine Angew. Math. 490, 55–64 (1997; Zbl 0882.22007)] where they made the stronger assumption that \(\mathcal C_0\) and \(\mathcal C_x\) are isomorphic for every point \(x \in X\). The main difficulty of the proof is that a priori there might exist a divisor \(H \subset X\) such that for \(x \in H\), the VMRT \(\mathcal C_x\) is not isomorphic to \(\mathcal C_0\). In order to exclude this possibility the author uses a notion of parallel transport along the tautological lifting of a standard minimal rational curve, a concept that already appeared implicitly in N. Mok [Trans. Am. Math. Soc. 354, No. 7, 2639–2658 (2002; Zbl 0998.32013)].

For the entire collection see [Zbl 1135.00009].

Reviewer: Andreas HĂ¶ring (Paris)